Usually, according to a practical problem, a mathematical model consists of several main parts: decision variables, environmental variables, objective functions and constraints. Decision variables represent the factors that decision makers can control, that is, controllable inputs, which are unknown variables in the model and need to be determined by model solution. Environmental variables represent the uncontrollable external factors of decision makers, that is, uncontrollable inputs, and their specific values need to be determined in the data collection stage and expressed as constants in the model.
The objective function refers to the mathematical equation that describes the objective of the problem, and the constraint condition refers to the mathematical expression (equality or inequality) that describes the restrictive factors in the problem.
Beginner mathematical modeling may sound a bit complicated, but it doesn't need to be as complicated as you think. In fact, we have been exposed to this process in junior high school and even primary school. The following is a topic that everyone will do in senior one. Enclose a rectangular field with a fence with a total length of 60 meters. The area s of a rectangular field varies with the length of one side of the rectangle. When x is what, the area s of the rectangular field changes.
This problem is a typical process of transforming a mathematical problem into a mathematical model. In fact, it is not much different from the mathematical modeling that high schools and even college students often participate in now, just for the convenience of programming and easy understanding. In mathematical modeling, we usually convert complex models into general forms, and the general forms of commonly used optimization models are as follows:
If the above problems are described in general form, they are as follows:
However, compared with this simple pure mathematical problem, mathematical modeling has two characteristics:
The (1) model is relatively complex.
In the mathematical modeling competition, because many problems come from practical problems, and even many problems will involve some mechanism problems, such as the furnace temperature curve of Question A in 2020 and the heat conduction model in the clothing design of 20 18 high temperature operation, the problems are relatively complex, so it is relatively difficult to model such practical problems.
It is usually difficult to model this complex problem in one step, and a gradual evolution method is needed. Start with a simple model (ignoring some difficult factors), and then gradually add more related factors to make the model evolve and make it closer to practical problems. The value of conclusions or suggestions based on model analysis is closely related to the degree to which the model describes the actual situation. Generally speaking, the closer the model is to reality, the greater the value of the analysis results.
Therefore, in the process of mathematical modeling, we should be able to tolerate the "distortion" of the model to the prototype to a certain extent. Learning to simplify the model properly is also a very important step in mathematical modeling.
(2) The solution is complex.
The pure mathematical problem mentioned above can be solved by a quadratic equation, but the actual problem solving is more complicated. This is why we mentioned that we should learn to simplify the model. In fact, the purpose of simplifying the model is to facilitate the solution. So maybe everyone has heard of all kinds of solving algorithms in mathematical modeling, which is also the difficulty of mathematical modeling.